Finite Group Databasebeta

← #10811985

Subgroups — #10811985

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
3#82462C211yesmaximal, characteristic, complemented
4#70486C212yesmaximal, complemented
5#70479C211yesmaximal, characteristic
6#70458C212yesmaximal
7#70456C211yesmaximal, characteristic, complemented
8#69773C212yesmaximal
9#69771C211yesmaximal, characteristic, complemented
10C4×#816C212yesmaximal, complemented
11C4×#821C211yesmaximal, characteristic
12#22541—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15#5584—24no—
16#5584C418yescomplemented
17C2×#898—22no—
18C2×#892—21no—
19C2×#875C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2212yes—
23C3×C8×Q8—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27C2×C3×C4×C8C2212yes—
28#5229—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2212yes—
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4980—22no—
37Dic3×C4⋊C4—21no—
38#4936C2212yes—
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41C4×C4⋊Dic3C2212yes—
42#4551C2212yes—
43#4551C2211yescharacteristic
44#4551C2211yescharacteristic
45C2×#556—31no—
46C2×C8×Q8D311yescharacteristic, complemented
47#2355—31no—
48#1983—32no—
49#1981—31no—
50#1958—32no—
51#1956—31no—
52#1805—32no—
53#1802—31no—
54C4×#103—32no—
55C4×#102—31no—
56C22×C3×Q8D411yescharacteristic
57C22×D12C2311yescharacteristic
58C22×C4×D3—21no—
59C22×Dic6C2312yes—
60C22×C3×C8C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×Q8C2×C414yes—
64C3×C4×Q8D414yes—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D412yes—
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2312yes—
69#946—28no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—22no—
85#898—216no—
86#892—28no—
87#875—24no—
88#875C2×C418yes—
89C4×D12—22no—
90C4×D12C2×C414yes—
91#869—22no—
92#869C2×C414yes—
93C4×Dic6—24no—
94C4×Dic6C2×C418yes—
95C3×C4⋊C8—24no—
96C3×C4⋊C8—22no—
97C3×C4×C8—24no—
98C3×C4×C8—22no—
99C3⋊#95—22no—
100C3⋊#95—21no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103#821—22no—
104#821—22no—
105#821—42no—
106#816—24no—
107#816—24no—
108#816—42no—
109#816—42no—
110#556—38no—
111#556—64no—
112C2×C4⋊Q8—31no—
113C2×#126—62no—
114C2×#124—32no—
115C2×C22⋊Q8—61no—
116C2×C4×Q8—32no—
117C2×C4×Q8C2×D311yescharacteristic
118C2×C4×D4—31no—
119C2×#117—61no—
120C8×Q8—24no—
121C2×C4⋊C8C2×D311yescharacteristic
122C2×C4⋊C8C2×D312yes—
123C2×#103—32no—
124C2×#103—32no—
125C2×#103—32no—
126C2×#103—32no—
127C2×#102—31no—
128C2×#102—31no—
129C2×#102—32no—
130C2×C4×C8C2×D311yescharacteristic
131C2×C4×C8C2×D312yes—
132#391—61no—
133#387—62no—
134#384—31no—
135#382—32no—
136C4×C4⋊C4—31no—
137C4×C4⋊C4—32no—
138C4×C4⋊C4—61no—
139#340—32no—
140#340—31no—
141#340—31no—
142C23×D3—21no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8C22⋊C414yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C2×D12—21no—
157C2×D12—21no—
158C2×D12—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12C22×C412yes—
162C2×C4×D3—24no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6C22×C414yes—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—21no—
175C2×C3×C8—22no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8#4112yes—
180C3×C4⋊C4C22⋊C414yes—
181C3×C4⋊C4C22⋊C418yes—
182C3×C42C22×C414yes—
183C3×C42C22×C412yes—
184C3×C42C2×D414yes—
185C3×C42C2×D412yes—
186(C2×D3)⋊C4—24no—
187(C2×D3)⋊C4—28no—
188(C2×D3)⋊C4—28no—
189C4⋊Dic3—24no—
190C4⋊Dic3—24no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—21no—
196C4⋊Dic3—22no—
197C4⋊Dic3—42no—
198C4⋊Dic3—81no—
199C4⋊Dic3C22×C412yes—
200C4⋊Dic3#4112yes—
201Dic3⋊C4—28no—
202C4×Dic3—28no—
203C4×Dic3—42no—
204C4×Dic3—82no—
205C22×Q8—32no—
206C22×Q8D1211yescharacteristic
207C22×D4—31no—
208C23×C4—61no—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D312yes—
211C22×C8C22×D311yescharacteristic
212C4⋊Q8—34no—
213C4⋊Q8—62no—
214#126—616no—
215#124—38no—
216#124—64no—
217C22⋊Q8—68no—
218C4×Q8—38no—
219C4×Q8—64no—
220C4×Q8C4×D314yes—
221C4×Q8#6714yes—
222C4×D4—34no—
223C4×D4—62no—
224#117—68no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4D1211yescharacteristic
235C2×C4⋊C4D1212yes—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—62no—
239C2×C42—62no—
240C2×C42—61no—
241C2×C42—62no—
242C2×C42C22×D311yescharacteristic
243C2×C42C22×D312yes—
244C4⋊C8—22no—
245C4⋊C8—24no—
246#103—64no—
247#103—64no—
248#103—122no—
249#103—122no—
250#102—62no—
251#102—62no—
252#102—122no—
253C4×C8—22no—
254C4×C8—24no—
255#95—61no—
256#95—62no—
257#95—62no—
258#95—62no—
259C23×C3C22×D411yescharacteristic
260C22×D3—22no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C3×Q8—28no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4C2×C22⋊C414yes—
275C2×C3×C4C2×C22⋊C414yes—
276C2×C3×C4C2×C22⋊C412yes—
277C2×C3×C4C2×C22⋊C412yes—
278C2×C3×C4#13512yes—
279C2×C3×C4#13512yes—
280C2×C3×C4C23×C411yescharacteristic
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4H(5,𝔽2)11yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×Dic3—22no—
287C2×Dic3—24no—
288C2×Dic3—22no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301D12—22no—
302D12—22no—
303D12—41no—
304D12—41no—
305C4×D3—48no—
306Dic6—24no—
307Dic6—24no—
308Dic6—42no—
309Dic6—42no—
310C3×C8—22no—
311C3×C8—42no—
312C3×C8—41no—
313C24—61no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—34no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×Q8—62no—
322C2×Q8(C2×D3)⋊C414yes—
323C2×D4—32no—
324C2×D4—61no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—61no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—61no—
334C22×C4—62no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C2×D1212yes—
338C22×C4C2×D1212yes—
339C22×C4C2×D1211yescharacteristic
340C22×C4C2×D1211yescharacteristic
341C22×C4C23×D311yescharacteristic
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8#23912yes—
351C4⋊C4—32no—
352C4⋊C4—32no—
353C4⋊C4—64no—
354C4⋊C4—68no—
355C4⋊C4—64no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—61no—
362C4⋊C4—122no—
363C4⋊C4—241no—
364C4⋊C4(C2×D3)⋊C414yes—
365C4⋊C4(C2×D3)⋊C418yes—
366C22⋊C4—68no—
367C22⋊C4—64no—
368C22⋊C4—68no—
369C42—68no—
370C42—122no—
371C42—242no—
372C42C2×C4×D312yes—
373C42C2×C4×D314yes—
374C42C2×#6712yes—
375C42C2×#6714yes—
376C22×C3C22×C22⋊C411yescharacteristic
377C22×C3#54811yescharacteristic
378C22×C3#55011yescharacteristic
379C22×C3C2×#13511yescharacteristic
380C22×C3C2×#13511yescharacteristic
381C22×C3#57711yescharacteristic
382C22×C3#57811yescharacteristic
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C2×D3—81no—
390C3×C4—24no—
391C3×C4—24no—
392C3×C4—22no—
393C3×C4#41614yes—
394C3×C4C4×#4111yescharacteristic
395C3×C4#52011yescharacteristic
396C3×C4#54811yescharacteristic
397C3×C4#55011yescharacteristic
398Dic3—42no—
399Dic3—44no—
400Dic3—42no—
401Dic3—81no—
402Dic3—82no—
403C23—62no—
404C23—121no—
405C23—121no—
406C23—121no—
407C23—121no—
408C23—121no—
409C23—121no—
410C23C22×D1211yescharacteristic
411Q8—28no—
412Q8—64no—
413Q8—64no—
414Q8—122no—
415Q8—122no—
416D4—62no—
417D4—62no—
418D4—121no—
419D4—121no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—62no—
427C2×C4—64no—
428C2×C4—62no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—128no—
436C2×C4—121no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4#90412yes—
443C2×C4#90412yes—
444C2×C4C2×(C2×D3)⋊C414yes—
445C2×C4C2×(C2×D3)⋊C412yes—
446C2×C4C2×(C2×D3)⋊C414yes—
447C2×C4C2×(C2×D3)⋊C412yes—
448C2×C4C22×C4×D311yescharacteristic
449C2×C4C2×#23911yescharacteristic
450C2×C4#100711yescharacteristic
451C2×C4D3×#4111yescharacteristic
452C2×C4#100911yescharacteristic
453C2×C4C22×#6711yescharacteristic
454C8—22no—
455C8—41no—
456C8—42no—
457C2×C3#234111yescharacteristic
458C2×C3C2×#41611yescharacteristic
459C2×C3#289811yescharacteristic
460C2×C3#336011yescharacteristic
461C2×C3#336411yescharacteristic
462C2×C3#339311yescharacteristic
463C2×C3#339411yescharacteristic
464D3—82no—
465C22—122no—
466C22—122no—
467C22—122no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22#557611yescharacteristic
473C22#557711yescharacteristic
474C22C2×#90411yescharacteristic
475C22C2×#90411yescharacteristic
476C22#575111yescharacteristic
477C22#575311yescharacteristic
478C22C22×(C2×D3)⋊C411yescharacteristic
479C4—24no—
480C4—24no—
481C4—22no—
482C4—124no—
483C4—122no—
484C4—122no—
485C4—241no—
486C4—242no—
487C4#511314yes—
488C4C4×#23911yescharacteristic
489C4#557411yescharacteristic
490C4#581711yescharacteristic
491C4#583411yescharacteristic
492C3#2254111yescharacteristic, complemented, Sylow
493C2—242no—
494C2#7991211yescharacteristic
495C2#7991311yescharacteristic
496C2#7992611yescharacteristic
497C2#7993411yescharacteristic
498C2#8245711yescharacteristic
499C2C2×#511311yescharacteristic
500C2#8380611yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C8×Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenterC2×C8×Q8D3
hypocenterC3#22541
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×C8×Q8C2semidirect
3. #82462C2semidirect
4. #70486C2semidirect
7. #70456C2semidirect
9. #69771C2semidirect
10. C4×#816C2semidirect
16. #5584C4semidirect
46. C2×C8×Q8D3semidirect
492. C3#22541semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55842
C2×C3×C8×Q82
#824622
#704862
#704792
#704582
#704562
#697732
#697712
C4×#8162
C4×#8212
#225413